# prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact

### Asked by muskanmahek2411 | 8th Oct, 2021, 10:48: PM

### Given: A circle C (O, r) and a tangent AB at a point P.
###

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

_{}OQ>OR

_{}OQ>OP (OR=OP=radius)

Thus, OP

But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.

Hence, OP is perpendicular to AB.

###

Given: A circle C (O, r) and a tangent AB at a point P.

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

_{}OQ>OR

_{}OQ>OP (OR=OP=radius)

Thus, OP

But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.

Hence, OP is perpendicular to AB.Gi

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

_{}OQ>OR

_{}OQ>OP (OR=OP=radius)

Thus, OP

But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.

Hence, OP is perpendicular to AB.

Given: A circle C (O, r) and a tangent AB at a point P.

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

_{}OQ>OR

_{}OQ>OP (OR=OP=radius)

Thus, OP

But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.

Hence, OP is perpendicular to AB.Gi

### Answered by Renu Varma | 23rd Oct, 2021, 06:28: PM

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